Let \(j\)  denote the iteration index in MSBL.
Stated in our notation, the overall estimation procedure is summarized in Algorithm REF .
The MSBL algorithm converges to a saddle point or a local maximizer of the overall log-likelihood [1]}.
Further, the MSBL algorithm has been empirically shown to correctly recover the support of \(\mathbf {Z}_t^k\) , provided \(\tau \)  and \(N\)  are large enough [1]}, if the signal to noise ratio is good enough.
The algorithm is run for \(j_{\max }\)  iterations in each of the \(T\)  REs.
As the iterations proceed, the hyperparameters corresponding to users with \(g_{ti} = 0\)  converge to zero, resulting in sparse estimates.
At the end of the iterations, the estimated coefficient \(\hat{g}_{tm}^k\)  for the \(m\) th user in the \(t\) th RE in the \(k\) th decoding iteration is obtained by thresholding \([{ \mathbf {\gamma }}_{kt}^{j_{\max }}]_m\)  at a value \(\gamma _{\text{pr}}\) .
This can result in errors in estimating \(g_{ti}\) , and the errors in APM estimation can be described by
\(\mathcal {F}_t^k & = \lbrace i \in [M^k] \ | \ \hat{g}_{ti}^k (1 - {g}_{ti}) = 1\rbrace , \\\mathcal {M}_t^k &= \lbrace i \in [M^k] \ | \ (1 - \hat{g}_{ti}^k) {g}_{ti} = 1\rbrace ,\) 
